Results 1 to 4 of 4

Thread: Inner Product Space

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    21

    Inner Product Space

    Let V be an inner product space. Let \{e_n\}_1^\infty be a complete orthonormal sequence in V. Suppose that f,g \in V with f = \sum_{n=1}^{\infty}c_ne_n and g= \sum_{k=1}^\infty d_ke_k. Prove that \langle f, g \rangle = \sum_{n=1}^\infty c_n \bar{d_n}.

    My proof so far:

    \langle f, g \rangle = \langle \sum_{n=1}^{\infty}c_ne_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \langle e_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \bar{d_n} \langle e_n, e_n \rangle

    I definitely don't think I can pull that sum out in the first step because the e_n has the sum with it as well. I was thinking of trying the following:

    \langle \lim_{n\to \infty} \sum_{n=1}^N c_n e_n , \lim_{k \to \infty} \sum_{k=1}^K d_k e_k \rangle

    I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by Benmath View Post
    Let V be an inner product space. Let \{e_n\}_1^\infty be a complete orthonormal sequence in V. Suppose that f,g \in V with f = \sum_{n=1}^{\infty}c_ne_n and g= \sum_{k=1}^\infty d_ke_k. Prove that \langle f, g \rangle = \sum_{n=1}^\infty c_n \bar{d_n}.

    My proof so far:

    \langle f, g \rangle = \langle \sum_{n=1}^{\infty}c_ne_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \langle e_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \bar{d_n} \langle e_n, e_n \rangle

    I definitely don't think I can pull that sum out in the first step because the e_n has the sum with it as well. I was thinking of trying the following:

    \langle \lim_{n\to \infty} \sum_{n=1}^N c_n e_n , \lim_{k \to \infty} \sum_{k=1}^K d_k e_k \rangle

    I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.
    What do you mean "pull out the limits"? Like \displaystyle \lim_{n\to\infty}\lim_{k\to\infty}\left\langle\sum  _{n=1}^{N}c_ne_n,\sum_{k=1}^{K}d_ke_k\right\rangle?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2008
    Posts
    21
    I definitely think that I cannot just pull out the sum like I did in the top, so I was trying to use partial sums, but I don't know where to go from there. Yes, what I mean was pulling out the limits like you have, but I don't even know if that would be helpful.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    There is no problem about pulling the sums out of the inner products, because the inner product is a continuous function of its two arguments. If f = \sum_{n=1}^{\infty}c_ne_n then f is the limit of the finite sums \sum_{n=1}^{N}c_ne_n. It follows that

    \langle f, g \rangle = \lim_{N\to\infty}\Bigl\langle \sum_{n=1}^{N}c_ne_n, g \Bigr\rangle =  \lim_{N\to\infty}\sum_{n=1}^{N}c_n\langle e_n,g\rangle = \sum_{n=1}^{\infty}c_n\langle e_n,g\rangle,

    and similarly for the right-hand term in the inner product.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: Aug 16th 2011, 02:52 PM
  2. Replies: 2
    Last Post: Apr 1st 2011, 02:40 AM
  3. Inner Product Space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Jan 23rd 2011, 07:28 PM
  4. closed in dual space E* implies closed in product space F^E
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 21st 2010, 04:58 AM
  5. inner product space
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Apr 5th 2010, 04:02 PM

Search Tags


/mathhelpforum @mathhelpforum